There are many instances in which it is desirable to predict the likelihood of an event occurring within a certain amount of time or the amount of time until an event is likely to occur. Indeed, event prediction type data, including censored data, is one of the most common types of data used in bioscience (for example). Predicting the occurrence of an event can help people plan for the occurrence of the event. For example, it is desirable to predict the time to recurrence of diseases or other health issues, such as cancer, or environmental events (e.g., earthquakes, hurricanes).
Censored data comprises incomplete data in which it is unknown when an event occurred/recurred. For example, in training a model to predict the recurrence of cancer in a patient, the training data would preferably include censored data comprising patient data in which no recurrence of cancer came about in particular patients. This is because most medical data includes both censored and non-censored data, and increasing the amount of data available to train a predictive model can increase the reliability and predictive power of the model. Censored data indicates whether the outcome under observation, e.g., recurrence of cancer, has occurred (for example) within a patient's follow-up visit time: if the recurrence of cancer has not been observed at a patient's follow-up visit, this patient's data is censored. In predicting recurrence of cancer (in patients who have been considered cured, for example), data for many patients may be censored. Such censored observation provides incomplete information about the outcome, since the event may eventually occur after the follow-up visit, which should be taken into account by a predictive model. However, the current most accurate learning models, particularly machine learning techniques involving neural networks and support vector machines, do not make use of such censored data.
It would be highly desirable when training a predictive model to have as much data from as many sources as possible. Thus, for example, for disease related events, it is generally desirable to have data from as many patients as possible, and as much data from each patient as possible. With such data, however, come difficulties in how to process censored data.
Typically, traditional survival analysis, e.g., the Cox proportional hazards model, uses censored data. However, in general, the reliability of the Cox model deteriorates if the number of features is greater than the number of events divided by 10 or 20 [1]. For example, in one study included as an example for the present invention, the dataset consisted of only 130 patients, each of which was represented by a vector of 25 features. For such data, the Cox model could not be successfully derived from this dataset until the feature dimensionality was reduced.
Neural networks have been shown to be able to outperform traditional statistical models, due to neural networks' capacity to model nonlinearities. However, in order to be successful, a neural network typically requires a large number of samples in the training set. Generally, several approaches have been used in applying survival data in neural networks. One approach is to model the hazard or survival function as a neural network structure. For example, constructing the survival curve by a hazard function modeled by a neural network, for which the ith output is the estimated hazard at the discretized time interval i. Others have used the discretized time interval as an additional input to a neural network to model the survival probability. Still others have used several separately trained networks, each used to model the hazard function at a different time interval.
Still, in order to effectively use machine learning algorithms, treatment of censored data is crucial. Simply omitting the censored observations or treating them as non-recurring samples bias the resulting model and, thus, should be avoided. Kaplan-Meier estimates of event probability have been used as target values during training for patients who had short follow-up times and did not have the event recurred. Although this algorithm takes into account, to some extent, both follow-up time and censoring, it still fails to make complete use of available information. For instance, it treats two recurred patients as the same regardless of their survival time.